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 A154244 a(n) = 6*a(n-1) - 2*a(n-2) for n>1; a(1)=1, a(2)=6. 8
 1, 6, 34, 192, 1084, 6120, 34552, 195072, 1101328, 6217824, 35104288, 198190080, 1118931904, 6317211264, 35665403776, 201358000128, 1136817193216, 6418187159040, 36235488567808, 204576557088768, 1154988365396992 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Binomial transform of A126473. lim_{n -> infinity} a(n)/a(n-1) = 3+sqrt(7) = 5.6457513110.... a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} avoiding 01 and 02. - Milan Janjic, Dec 17 2015 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..500 Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e). Index entries for linear recurrences with constant coefficients, signature (6,-2). FORMULA a(n) = ((3 + sqrt(7))^n - (3 - sqrt(7))^n)/(2*sqrt(7)). G.f.: x/(1-6*x+2*x^2). - Philippe Deléham, Jan 06 2009 MATHEMATICA a[n_]:=(MatrixPower[{{1, 3}, {1, 5}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *) LinearRecurrence[{6, -2}, {1, 6}, 40] (* Vincenzo Librandi, Feb 02 2012 *) PROG (MAGMA) Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((3+r)^n-(3-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009 (MAGMA) I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 02 2012 (Sage) [lucas_number1(n, 6, 2) for n in xrange(1, 22)] # Zerinvary Lajos, Apr 22 2009 (Maxima) a[1]:1\$ a[2]:6\$ a[n]:=6*a[n-1]-2*a[n-2]\$ makelist(a[n], n, 1, 21);  // Bruno Berselli, May 30 2011 (PARI) Vec(1/(1-6*x+2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 28 2011 CROSSREFS Equals 1 followed by 2*A010913 (Pisot sequence E(3,17)). Cf. A010465 (decimal expansion of square root of 7), A126473. Sequence in context: A052264 A049608 A244937 * A273583 A126501 A218990 Adjacent sequences:  A154241 A154242 A154243 * A154245 A154246 A154247 KEYWORD nonn,easy AUTHOR Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009 EXTENSIONS Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009 Edited by Klaus Brockhaus, Oct 06 2009 Name (corrected) from Philippe Deléham, Jan 06 2009 STATUS approved

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